Two meanings of “if”? Individual differences in the interpretation of conditionals

Abstract

This work investigates the nature of two distinct response patterns in a probabilistic truth table evaluation task, in which people estimate the probability of a conditional on the basis of frequencies of the truth table cases. The conditional-probability pattern reflects an interpretation of conditionals as expressing a conditional probability. The conjunctive pattern suggests that some people treat conditionals as conjunctions, in line with a prediction of the mental-model theory. Experiments 1 and 2 rule out two alternative explanations of the conjunctive pattern. It does not arise from people believing that at least one case matching the conjunction of antecedent and consequent must exist for a conditional to be true, and it does not arise from people adding the converse to the given conditional. Experiment 3 establishes that people's response patterns in the probabilistic truth table task are very consistent across different conditionals, and that the two response patterns generalize to conditionals with negated antecedents and consequents. Individual differences in rating the probability of a conditional were loosely correlated with corresponding response patterns in a classical truth table evaluation task, but there was little association with people's evaluation of deductive inferences from conditionals as premises. A theoretical framework is proposed that integrates elements from the conditional-probability view with the theory of mental models.

This research was supported by Deutsche Forschungsgemeinschaft (DFG, Grant FOR 375 1–1). We thank Annekatrin Hudjetz and Moritz Ischebek for help in programming the experiments. We are especially indebted to Mirko Wendland for posting our experiments on his W-Lab server, http://w-lab.de, and to Ulf Reips for broadly advertising them through his web page, http://www.psychologie.unizh.ch/sowi/Ulf/Lab/WebExpPsyLabD.html

Notes

¹The number is smaller than the 196 participants in the lower left cell of Table 2, because the latter also included participants with ratio > 40 and frequency < –40, which we now reclassified as those who understand the conditional as material.

²There are four basic inference patterns involving a conditional “If p then q” as premise. Modus ponens (MP) is obtained by adding a minor premise “p”, which licenses the inference that “q”. Modus tollens (MT) has “not q” as minor premise and licenses the conclusion that “not p”. Denial of the antecedent (DA) has the minor premise “not p” and the associated conclusion “not q”, whereas acceptance of the consequent (AC) works with the minor premise “q” and the conclusion “p”. DA and AC are not valid according to formal logic, but nonetheless are often endorsed by logically untrained people.

³Johnson-Laird and Byrne (1991) argued that a representation consisting of a single TT model can be used to explain the TFII pattern: The TT case matches the explicit model and therefore is judged T, and the false-antecedent cases are not explicitly modelled and are therefore regarded as irrelevant. But how can people figure out that the TF case renders the conditional false? In the version of the model theory presented in 1991, the single model of the TT case was augmented with symbolic annotations that served this purpose. A model of “If p then q” was noted as [p] q, where the square brackets indicated that the antecedent term was represented exhaustively. Therefore, no further model could involve p, and this rules out the p & ¬q case as a possibility. The more recent version of the model theory (Johnson-Laird & Byrne, 2002), however, dropped the square brackets and thereby undermined the explanation for the TFII pattern in the classical truth table task. The theory must predict pattern TFFF or pattern TIII from the single-model representation.

⁴For the general case, including conditionals with negations, the frame represents the possible cases in which the antecedent is true, and the mental model represents the conjunction of true antecedent and true consequent. We use the conditional without negation in the discussion of our account for ease of exposition.